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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a> or <font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V3">non-decreasing</a> )
 </span><b>by </b><i><a class="txt" href="setlim_2.html#E1:28"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/><div class="add"><a NAME="E2:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 <b>by </b><i><a class="ref" href="setlim_2.html#T24" target="_self">Th24</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 <b>by </b><i><a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/><div class="add"><a NAME="E2:28_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="prob_1.html#V3">non-decreasing</a>
 <b>by </b><i><a class="ref" href="setlim_2.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E3:28_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="setlim_2.html#K6" target="_self">(\/)</a> <font color="Maroon">c<sub>3</sub></font> is <a href="setlim_1.html#V1">monotone</a>
 <b>by </b><i><a class="ref" href="setlim_1.html#D1">SETLIM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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